Asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters. II.
In the present study the authors deal with the ground states of the quasilinear elliptic equation \[ -\Delta_mu=-\delta u^{m-1}+u^{p-1} \text{ in }\mathbb{R}^N\tag{1} \] in which \(1<m<N\), \(m<p<m^*\), \(\delta> 0\) and \(m^*\) is the critical Sobolev exponent \(m^*=\frac{Nm}{N-m}\). Here by \(\Delta_m\) is denoted \(\Delta_mu:=\text{div}(|\nabla u|^{m-2}\nabla u)\). The authors mainly interested in a radially symmetric solutions for (1). The authors present the exact asymptotic behaviour of the ground state, for any \(1<m<N\), both at the origin and outside the origin, when the equation tends to critical growth. Moreover, the authors obtain the ``equilibrium'' relation between \(\varepsilon(p=m^*-\varepsilon)\) and \(\delta\) so that when they both vanish according to this relation, ground states neither blow up nor vanish. Part I, cf. ibid. 19, No. 4, 477--504 (2002; Zbl 1013.35031), Part III, J. Differ. Equations 198, No. 1, 53--90 (2004; Zbl 1046.35043).
scientific article; zbMATH DE number 2009152
scientific article